Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges

Abstract

We consider the planar orthogonal polynomial $p_n(z)$ with respect to the measure supported on the whole complex plane $${\mathrm{e}}^{-N|z{|}^2}\prod _{j=1}^{\nu }|z-a_j{|}^{2c_j}\mathrm{d}A(z)$$ where $\mathrm{d}A$ is the Lebesgue measure of the plane, $N$ is a positive constant, $\{c_1,\cdots ,c_{\nu }\}$ are nonzero real numbers greater than $-1$ and $\{a_1,\cdots ,a_{\nu }\}\subset \mathbb{D}\setminus \{0\}$ are distinct points inside the unit disk. In the scaling limit when $n/N=1$ and $n\to \infty$ we obtain the strong asymptotics of the polynomial $p_n(z)$. We show that the support of the roots converges to what we call the "multiple Szego curve," a certain connected curve having $\nu +1$ components in its complement. We apply the nonlinear steepest descent method on the matrix Riemann-Hilbert problem of size $(\nu +1)\times (\nu +1)$.

arXiv comment61 pages, 12 figures. In the revised version, we correct one phase factor see Definition 1.4

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• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• Orthogonal polynomials in the normal matrix model with two insertions
• The Schwarz function and the shrinking of the Szegő curve: electrostatic, hydrodynamic, and random matrix models